Some time ago, I said two of the things I would like to have in a PSK are a measuring tape and a set of sine tables. <br><br>I decided that a table of tangents would make more sense, so I generated such a table in Excel. (Described below for anyone who wants it.)<br><br>To use the table, set up a vertical stick and monitor the shadow it casts. At the equinox, the angle the sun makes with the stick should be equal to your latitude. <br><br>To compute this, measure the length of the shadow at its shortest point and divide it by the height of the stick. Obviously, the ground has to be level and the stick has to be absolutely vertical (or you can use a plumb line). This value will be the tangent of your latitude; find the closest value in the table and read off the corresponding degrees and minutes.<br><br>For example, supposing the shadow cast at noon is exactly 1 foot in length and the stick is exactly 3 feet high. Divide 1 by 3 to get 1/3, or .33333...<br><br>The closest value in the table is 0.33; the corresponding entry in my table is <br><br>0.33 18.26288994 18 16<br><br>giving a latitude of 18 degrees, 16 minutes.<br><br>My table only goes to 45 degrees, but that's no problem. Above 45 degrees latitude, the shadow will be longer than the stick. Simply divide the height of the stick by the length of the shadow, find the value as before, and subtract it from 90 degrees.<br><br>For example, suppose the 3-foot high stick casts a 4-foot shadow at its shortest point. 3 divided by 4 equals .75, for which the entry is<br><br>0.75 36.86989765 36 52<br><br>giving a value of 36 degrees 52 minutes. We have to subtract this from 90 degrees, which gives us a latitude of 53 degrees, 8 minutes.<br><br>Of course, this is only valid, strictly speaking, at the spring or autumn equinox (i.e. when the sun is directly over the equator at noon). But if you can relay your position to Search and Rescue by radio, they should be able to apply the correction and at least have a good idea where to start looking. And the mental exercise will keep you warm :-)<br><br>The Excel Spreadsheet<br><br>After some thought, I realized the simplest way to generate this table would be to list all the values from 0 to 1, in increments of .01, and take the arctan (inverse tangent) for each.<br><br>In cell A1, write "Tangent". In cell A2, write "0.01". (There's no need to write down the tangent for 0; it's zero degrees.) In cell A3, write "=a2+.01".<br><br>In cell B2, write "=degrees(atan(a2))".<br><br>In cell C1, write "Degrees". In cell C2, write "=trunc(b2)"; ;this gives you the integer portion of b2, which is the number of whole degrees.<br><br>In cell D1, write "Minutes". In cell D2, write "=round(60*(b2-c2),0)". This gives you the number of minutes, rounded off to the nearest whole number.<br><br>Now select the block (a3..d100) and press Ctrl-D (for Fill Down). The resulting table should look something like this:<br><br>Tangent Degrees Degrees Minutes<br>0.01 0.572938698 0 34<br>0.02 1.145762838 1 9<br>0.03 1.718358002 1 43<br>0.04 2.290610043 2 17<br>0.05 2.862405226 2 52<br>0.06 3.433630362 3 26<br>0.07 4.004172941 4 0<br>0.08 4.57392126 4 34<br>.....

Huh? :-)

Hey jw, didn't you know that "geometry" means "earth measuring"? And that it was invented by the ancient Greeks to help them navigate? :-)

Your tables brought back warm and fuzzy memories of using the sextant,which, usually left my brain warm and fuzzy. The prehistory and history of navigation is fascinating. I do question our utter reliance on systems understood by fewer and fewer users. Knowing the changing temperature of tides and the habits of birds are still of value ;O)

I did some research on the calculation of the declination (the difference between the angle of the sun and the latitude). I found approximations for that value. Based on that information I wrote the following function for StarOffice:<br><br>Function Declination (cd)<br> ref= datevalue ("1.1."+year(cd))<br> longitude = 0<br> t = cd -ref + 1 + (12 - longitude / 15 ) / 24<br> M = (0.9856 * t - 3.289)<br> L = (M + 1.916 * sin((M*Pi/180)) + 0.02 * sin((2 * M)*Pi/180) + 282.634)<br> sindec = 0.39782 * sin(L*Pi/180)<br> Declination = atn(sindec/sqr(1 - sindec^2))*180/PI<br> <br>End Function<br><br>cd is a date,<br>cd - ref is the the day of the year, <br>Declination is in degrees<br><br> I used the function to create a graph.

LOL<br><br>jw's response was not, shall we say, entirely unexpected :-)<br><br>I do find it curious, though, that people who take pride in learning how to generate fire using quartz and fool's gold seem to balk at learning how to navigate using equally primitive methods. ;-)<br><br>For one thing, there seems to be a fashion in "survival" today that emphasizes the native American skills, probably as a result of books by authors like Tom Brown and Larry Dean Olsen. Not that there's anything wrong with that; but being of European descent myself (although not Greek), I think there's probably a lot we can learn from my ancestors as well. :-)<br><br>For another, mathematics (as taught in schools today) is insufferably boring - and I say this as someone who loves solving math puzzles. Would our children be more interested if we took them out in the woods for a weekend, broke them into groups of four, and challenged them to find their latitude and longitude using the mathematical skills they had learned in geometry and trigonometry? <br><br>Who can say?<br>

Wow! That's really neat.<br><br>I will definitely have to study that more closely.

If you haul around the tape and make stakes you can also effectively make use of the small angle equation:<br>http://ceres.hsc.edu/homepages/classes/astronomy/spring99/Mathematics/sec7.html<br><br>http://www.astro.washington.edu/astro211/smanAA/smangleA.html<br><br>http://dilbert.physast.uga.edu/~derek/ASTR1020/numbers.html<br><br>Of course, if you happen to have a topo map and can determine a height then you can accurately compute the distance...

Can't I just use the altitude off my GPS? Or is that not accurate enough? ;-)<br><br>I did dig out a textbook on Celestial Navigation, but it concentrates mainly on how to read an almanac, something I don't intend to cram into my PSK. However, I understood it far better than I did the last time I put it down. <br><br>It's been far too long since I played around with this stuff. Trying to distil the necessary formulae down to a piece of paper the size of an Altoid's tin is going to be a bit of a challenge :-)

"Hey jw, didn't you know that "geometry" means "earth measuring"? And that it was invented by the ancient Greeks to help them navigate? :-)"<br><br>Yes, and Yes. But the last thing I want to do in an emergency situtation is pull out a table and try to find my direction/inclination/declination/altitude/whatever using it. You can simply use a stick, a watch, a piece of string, etc. to do the exact same thing without using math. Very interesting thread though, something I definately want to explore!<br><br>John McIntire